Tarea 13

Ejercicio 1

Comprimir una imagen en blanco y negro usando PCA.

library(png)
tere <- readPNG('Image.png')
image(tere[,,1])

tereMat <- (tere[,,1]+tere[,,2]+tere[,,3])/3
dim(tereMat)

## [1] 960 720

# svdTere <- svd(tereMat)
# 
# #Con SVD
# for(k in c(10,30,60,100)){
#   print(k)
#   image(svdTere$u[,1:k] %*% diag(svdTere$d)[1:k,1:k] %*% t(svdTere$v)[1:k,], axes = FALSE, col = grey(seq(0, 1, length = 256)))
# }

#Con PCA
calculaImagenReducida <- function(img, k) {
  Data_mean = apply(img, 2, mean) #Saca la media de cada columna
  a <- dim(img) #Dimensiones de la imagen original
  Data_meanNew <- rep(Data_mean, a[1])
  DataAdjust = img - Data_meanNew
  cov_data = cov(DataAdjust)
  eig <- eigen(cov_data)
  V_trans = t(eig$vectors)
  DataAdjust_trans = t(DataAdjust)
  FinalData = V_trans %*% DataAdjust_trans
  #PCs = a[2] - k
  Reduced_V = eig$vectors[,1:k]
  Y=t(Reduced_V) %*% DataAdjust_trans
  Compressed_Data=Reduced_V %*% Y                                         
  Compressed_Data = t(Compressed_Data) + Data_meanNew
  Compressed_Data
}

# for(i in c(2,10,30,50,100,300)){
#   print(i)
#   image(calculaImagenReducida(tereMat, i), axes = FALSE, col = grey(seq(0, 1, length = 256)))
# }

tati <- readPNG('Tat.png')
tatMean <- (tati[,,1]+tati[,,2]+tati[,,3])/3
dim(tatMean)

## [1] 960 960

image(tatMean, axes = FALSE, col = grey(seq(0, 1, length = 256)), main='Original')

for(i in c(10,50,100,350, 500)){
  print(i)
  image(calculaImagenReducida(tatMean, i), , axes = FALSE, col = grey(seq(0, 1, length = 256)), main=paste(i, 'componentes'))
}

## [1] 10

## [1] 50

## [1] 100

## [1] 350

## [1] 500

Ejercicio 2

Implementa whitening en los datos faithful, compara las gráficas de los datos crudos y preprocesados.

Dats originales:

library(knitr)
kable(head(faithful))

eruptions	waiting
3.600	79
1.800	54
3.333	74
2.283	62
4.533	85
2.883	55

Media y matriz de covarianzas:

sapply(faithful, mean)

## eruptions   waiting 
##  3.487783 70.897059

cov(faithful)

##           eruptions   waiting
## eruptions  1.302728  13.97781
## waiting   13.977808 184.82331

Cov <- cov(faithful)
eig2 <- eigen(Cov)
E <- eig2$vectors
D <- diag(eig2$values)
Dinv2 <- diag(1/sqrt(eig2$values))
faithfulcent <- faithful - cbind(rep(sapply(faithful, mean)[1], nrow(faithful)), rep(sapply(faithful, mean)[2], nrow(faithful)))
WhitenedFaithful <- t(Dinv2 %*% t(E) %*% t(faithfulcent))

Los datos con whitening:

kable(as.data.frame(head(WhitenedFaithful)))

V1	V2
0.5932500	1.0117128
-1.2451557	0.8236565
0.2260839	0.7864494
-0.6573809	1.0714867
1.0372421	0.0459461
-1.1660200	-1.2087815

Media y matriz de covarianzas:

apply(WhitenedFaithful, 2, mean)

## [1] 4.265339e-16 6.873650e-16

cov(WhitenedFaithful)

##               [,1]          [,2]
## [1,]  1.000000e+00 -5.670137e-17
## [2,] -5.670137e-17  1.000000e+00

library(ggplot2)
ggplot(faithful) + geom_point(aes(x=eruptions, y=waiting)) + ggtitle('Originales')

ggplot(as.data.frame(WhitenedFaithful)) + geom_point(aes(x=V1, y=V2)) + ggtitle('Transformados')

Ejercicio 3

3. Realiza un mapa de un estado, grafica alguna variable de tu interés (índice de marginación, población, …) a nivel municipio coloreando los polígonos de acuerdo a esta variables. Si lo deseas utiliza como fondo una imagen de google maps.

library(rgdal)

## Loading required package: sp
## rgdal: version: 0.9-1, (SVN revision 518)
## Geospatial Data Abstraction Library extensions to R successfully loaded
## Loaded GDAL runtime: GDAL 1.10.1, released 2013/08/26
## Path to GDAL shared files: /usr/share/gdal/1.10
## Loaded PROJ.4 runtime: Rel. 4.8.0, 6 March 2012, [PJ_VERSION: 480]
## Path to PROJ.4 shared files: (autodetected)

library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'
## 
## The following object is masked from 'package:stats':
## 
##     filter
## 
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

pob <- read.csv('pobl.csv')
mun_shp <- readOGR("./municipios" , "MUNICIPIOS")

## OGR data source with driver: ESRI Shapefile 
## Source: "./municipios", layer: "MUNICIPIOS"
## with 2463 features and 7 fields
## Feature type: wkbPolygon with 2 dimensions

mun_shp@data$id = rownames(mun_shp@data)
DF_shp <- subset(mun_shp, CVE_ENT == '09')

DF_df <- DF_shp %>% 
  fortify(region = "CVE_MUN") %>%
  arrange(order)

## Loading required package: rgeos
## rgeos version: 0.3-8, (SVN revision 460)
##  GEOS runtime version: 3.4.2-CAPI-1.8.2 r3921 
##  Polygon checking: TRUE

DF_df$id <- as.integer(DF_df$id)

DF_ind <- DF_df %>%
  mutate(CVE = id) %>%
  left_join(pob[,c('id','pob')])

## Joining by: "id"

ggplot() + 
  geom_polygon(data = DF_ind, aes(long, lat, group = group, fill = pob)) +
  labs(title = "Población en el DF", fill = "Población") +
  coord_fixed()